Andromeda
Note

Trigonometric Identities

Definition

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable.

  • Fundamental Identity (Theorem 11.2.1): sin⁡2α+cos⁡2α=1\sin^2 \alpha + \cos^2 \alpha = 1
  • Quotient Identity: tan⁡α=sin⁡αcos⁡α\tan \alpha = \frac{\sin \alpha}{\cos \alpha}
  • Reciprocal: csc⁡α=1sin⁡α\csc \alpha = \frac{1}{\sin \alpha}, sec⁡α=1cos⁡α\sec \alpha = \frac{1}{\cos \alpha}, cot⁡α=1tan⁡α\cot \alpha = \frac{1}{\tan \alpha}

Why It Matters

Identities are the ‘laws of conservation’ for angles. They establish the unbreakable relationships between trig functions, allowing for the deep structural transformations of equations that are necessary for solving complex integrals and wave equations.

Core Concepts

  • Fundamental Pythagorean Identity sin⁡2α+cos⁡2α=1\sin^2 \alpha + \cos^2 \alpha = 1

    • How to read: “Sine squared alpha plus cosine squared alpha equals one.”
    • Meaning: See full treatment in Pythagorean Identities. This is the root of almost every trig simplification. You will use the rearranged forms 1−sin⁡2=cos⁡21 - \sin^2 = \cos^2 and 1−cos⁡2=sin⁡21 - \cos^2 = \sin^2 constantly.

  • Quotient Identity tan⁡α=sin⁡αcos⁡α\tan \alpha = \frac{\sin \alpha}{\cos \alpha}

    • How to read: “Tan alpha equals sine alpha over cosine alpha.”
    • Meaning: Tan is literally the ratio of opposite over adjacent in a right triangle, which is (opp/hyp) / (adj/hyp) = sin/cos. Use when you need to convert between tan and the basic sin/cos pair (very common in integrals and identities).

  • Reciprocal Identities csc⁡α=1sin⁡α,sec⁡α=1cos⁡α,cot⁡α=1tan⁡α\csc \alpha = \frac{1}{\sin \alpha}, \quad \sec \alpha = \frac{1}{\cos \alpha}, \quad \cot \alpha = \frac{1}{\tan \alpha}

    • How to read: “Csc alpha equals one over sine alpha”, “Sec equals one over cos”, “Cot equals one over tan.”
    • Meaning: These are just the definitions of the co-functions. Memorize the prefixes: co-secant is reciprocal of sine, etc. They let you rewrite everything in terms of sin and cos when an identity or equation mixes all six functions. Note that cot⁡=cos⁡/sin⁡\cot = \cos/\sin is also true (equivalent form).

  • Complementary (co-function) identities

    • For α+β=90∘\alpha + \beta = 90^\circ: sin⁡α=cos⁡β\sin \alpha = \cos \beta, cos⁡α=sin⁡β\cos \alpha = \sin \beta, tan⁡α=cot⁡β\tan \alpha = \cot \beta, etc.
    • How to read: “Sine alpha equals cosine beta; alpha plus beta equals ninety degrees.”
    • Meaning: Complementary angles swap sine and cosine roles; special cases of the sum/difference formulas.

I have now processed a good first batch of high-density formula notes that did not have “formula” in their names.

Next, I should continue with more from the clusters (taylor general, graphs, conics, algebra, etc.), and also start hitting calculus derivative/integral specific notes.

To keep momentum, let’s read and enhance a few more important ones now.

Connected Concepts

Connected notes